Related papers: A density problem for Sobolev spaces on Gromov hyp…

We prove that for a bounded simply connected domain $\Omega\subset \mathbb R^2$, the Sobolev space $W^{1,\,\infty}(\Omega)$ is dense in $W^{1,\,p}(\Omega)$ for any $1\le p<\infty$. Moreover, we show that if $\Omega$ is Jordan, then…

We show that in a bounded Gromov hyperbolic domain $\Omega$ smooth functions with bounded derivatives $C^\infty(\Omega)\cap W^{k,\infty}(\Omega)$ are dense in the homogeneous Sobolev spaces $L^{k,p}(\Omega)$.

We show that in a bounded simply connected planar domain $\Omega$ the smooth Sobolev functions $W^{k,\infty}(\Omega)\cap C^\infty(\Omega)$ are dense in the homogeneous Sobolev spaces $L^{k,p}(\Omega)$.

We prove that for a domain $\Omega$ in a PI space $X$ such that $\Omega$ satisfies the Gehring Hayman condition and the ball separation condition, the Newtonian Sobolev space $N^{1,\infty}(\Omega)$ is dense in the space $N^{1,p}(\Omega)$…

We describe a class of Sobolev $W^k_p$-extension domains $\Omega\subset R^n$ determined by a certain inner subhyperbolic metric in $\Omega$. This enables us to characterize finitely connected Sobolev $W^1_p$-extension domains in $R^2$ for…

We show the density of smooth Sobolev functions $W^{k,\infty}(\Omega)\cap C^\infty(\Omega)$ in the Orlicz-Sobolev spaces $L^{k,\Psi}(\Omega)$ for bounded simply connected planar domains $\Omega$ and doubling Young functions $\Psi$.

We characterize bounded simply-connected planar $W^{1,p}$-extension domains for $1 < p <2$ as those bounded domains $\Omega \subset \mathbb R^2$ for which any two points $z_1,z_2 \in \mathbb R^2 \setminus \Omega$ can be connected with a…

We show that Sobolev space $W^1_1(\Omega)$ of any planar one-connected domain $\Omega$ has the Bounded Approximation property. The result holds independently from the properties of the boundary of $\Omega$. The prove is based on a new…

Let $\Omega$ be a smooth bounded domain in ${\mathbb R}^n$, $0\textless{}s\textless{}\infty$ and $1\le p\textless{}\infty$. We prove that $C^\infty(\overline\Omega\, ; {\mathbb S}^1)$ is dense in $W^{s,p}(\Omega ; {\mathbb S}^1)$ except…

In this paper we study the global geometry of the Kobayashi metric on "convex" sets. We provide new examples of non-Gromov hyperbolic domains in $\mathbb{C}^n$ of many kinds: pseudoconvex and non-pseudocon \newline -vex, bounded and…

For each $m\ge 1$ and $p>2$ we characterize bounded simply connected Sobolev $L^m_p$-extension domains $\Omega\subset R^2$. Our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in $\Omega$. Its proof is based on a…

In this paper we establish Gehring-Hayman type theorems for some complex domains. Suppose that $\Omega\subset \mathbb{C}^n$ is a bounded $m$-convex domain with Dini-smooth boundary, or a bounded strongly pseudoconvex domain with…

We consider the problem of strong density of smooth maps in the Sobolev space $ W^{s,p}(Q^{m};\mathcal{N}) $, where $ 0 < s < +\infty $, $ 1 \leq p < +\infty $, $ Q^{m} $ is the unit cube in $ \mathbb{R}^{m} $, and $ \mathcal{N} $ is a…

We describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $W^{s,p}(\Omega)$ for an open, bounded set $\Omega\subset\mathbb{R}^{d}$. The density property…

Let $\Omega$ be a domain in $\mathbb{C}$ with hyperbolic metric $\lambda_\Omega(z)|dz|$ of Gaussian curvature $-4.$ Mejia and Minda proved in their 1990 paper that $\Omega$ is (Euclidean) convex if and only if…

We investigate two density questions for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) for an open set $\Omega\subset\mathbb R^n$,…

We focus on the Sobolev spaces of bounded subanalytic submanifolds of $\mathbb{R}^n$. We prove that if $M$ is such a manifold then the space $\mathscr{C}_0^\infty(M)$ is dense in $W^{1,p}(M,\partial M)$ (the kernel of the trace operator)…

In this paper, we study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, we obtain the following results. 1- Let $1\leq q\leq p\leq \infty$. Then a bounded $(L^{1, p}, L^{1,…

In this paper we prove necessary and sufficient conditions for the Kobayashi metric on a convex domain to be Gromov hyperbolic. In particular we show that for convex domains with $C^\infty$ boundary being of finite type in the sense of…

We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic…